From (3) By Definition 1, is spanned by some linearly independent vectors with Hence with some depending on and This shows that is spanned by If we prove linear independence of these vectors, (1) will follow.

Suppose with some Then This tells us that Since the vector also belongs to by Exercise 3 it is zero. By linear independence, is possible only when This proves the linear independence of and the statement.

Exercise 2 (rank-nullity theorem) If is then

Proof. Exercise 4 and equation (2) imply But we know from Exercise 1 that